Properties

Label 1800.1.l.b
Level 1800
Weight 1
Character orbit 1800.l
Analytic conductor 0.898
Analytic rank 0
Dimension 2
Projective image \(S_{4}\)
CM/RM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 1800.l (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.10800.2
Artin image size \(48\)
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.34992000000.5

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{7}\) \(+O(q^{10})\) \( q\) \(+ q^{7}\) \( -\beta q^{11} \) \(- q^{13}\) \( -\beta q^{17} \) \(+ q^{19}\) \( + \beta q^{23} \) \( -\beta q^{29} \) \(+ q^{31}\) \(- q^{43}\) \( + \beta q^{47} \) \( + \beta q^{59} \) \(+ q^{61}\) \(+ q^{67}\) \( -\beta q^{77} \) \( -\beta q^{83} \) \(- q^{91}\) \(- q^{97}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 2q^{43} \) \(\mathstrut +\mathstrut 2q^{61} \) \(\mathstrut +\mathstrut 2q^{67} \) \(\mathstrut -\mathstrut 2q^{91} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
1.41421i
1.41421i
0 0 0 0 0 1.00000 0 0 0
1601.2 0 0 0 0 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7} \) \(\mathstrut -\mathstrut 1 \) acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\).